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%I #23 May 26 2019 14:55:11
%S 1,1,1,1,1,1,1,1,2,1,1,1,1,6,1,1,1,1,2,24,1,1,1,1,1,4,120,1,1,1,1,1,2,
%T 12,720,1,1,1,1,1,1,4,36,5040,1,1,1,1,1,1,2,8,144,40320,1,1,1,1,1,1,1,
%U 4,24,576,362880,1,1,1,1,1,1,1,2,8,72,2880,3628800,1
%N Number A(n,k) of permutations p of [n] such that p(i)-i is a multiple of k for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A275062/b275062.txt">Antidiagonals n = 0..140, flattened</a>
%F A(n,k) = Product_{i=0..k-1} floor((n+i)/k)!.
%F A(k*n,k) = (n!)^k = A225816(k,n).
%F For k > 0, A(n, k) ~ (2*Pi*n)^((k - 1)/2) * n! / k^(n + k/2). - _Vaclav Kotesovec_, Oct 02 2018
%e A(5,0) = A(5,5) = 1: 12345.
%e A(5,1) = 5! = 120: all permutations of {1,2,3,4,5}.
%e A(5,2) = 12: 12345, 12543, 14325, 14523, 32145, 32541, 34125, 34521, 52143, 52341, 54123, 54321.
%e A(5,3) = 4: 12345, 15342, 42315, 45312.
%e A(5,4) = 2: 12345, 52341.
%e A(7,4) = 8: 1234567, 1274563, 1634527, 1674523, 5234167, 5274163, 5634127, 5674123.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 24, 4, 2, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 120, 12, 4, 2, 1, 1, 1, 1, 1, 1, ...
%e 1, 720, 36, 8, 4, 2, 1, 1, 1, 1, 1, ...
%e 1, 5040, 144, 24, 8, 4, 2, 1, 1, 1, 1, ...
%e 1, 40320, 576, 72, 16, 8, 4, 2, 1, 1, 1, ...
%e 1, 362880, 2880, 216, 48, 16, 8, 4, 2, 1, 1, ...
%e 1, 3628800, 14400, 864, 144, 32, 16, 8, 4, 2, 1, ...
%p A:= (n, k)-> mul(floor((n+i)/k)!, i=0..k-1):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A[n_, k_] := Product[Floor[(n+i)/k]!, {i, 0, k-1}];
%t Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 26 2019, from Maple *)
%Y Columns k=0-10 give: A000012, A000142, A010551, A264557, A264635, A264656, A264701, A264791, A275063, A275064, A275065.
%Y A(k*n,n) for k=1..4 give: A000012, A000079, A000400, A009968.
%Y Cf. A225816.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Jul 15 2016
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